A293507 Expansion of e.g.f. exp(x/(1 - x^4)).

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 378001, 5473441, 39972241, 199679041, 7005552841, 176899522801, 2186722497961, 17454339826561, 459473703430561, 16503993702423361, 306140370496394401, 3555223271216311681, 80917223353652470681, 3568770455830785208081

Offset

0,6

Links

Table of n, a(n) for n=0..22.

Formula

E.g.f.: Product_{k>0} exp(x^(4*k-3)).

a(n) ~ exp(1/4 + sqrt(n) - n) * n^(n-1/4) / 2. - Vaclav Kotesovec, Oct 11 2017

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * (4*k+1)! * a(n-4*k-1). - Ilya Gutkovskiy, Feb 24 2022

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,k)/(n-4*k)!. - Seiichi Manyama, Jun 08 2024

Mathematica

CoefficientList[Series[E^(x/(1 - x^4)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 11 2017 *)

Prog

(PARI) N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1-x^4))))
(PARI) N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(4*k-3)))))

Crossrefs

E.g.f.: exp(x/(1 - x^m)): A000262 (m=1), A088009 (m=2), A293493 (m=3), this sequence (m=4).

Cf. A293526.

Sequence in context: A354554 A367720 A293566 * A356630 A306452 A238250

Adjacent sequences: A293504 A293505 A293506 * A293508 A293509 A293510

Keyword

nonn

Author

Seiichi Manyama, Oct 10 2017

Status

approved